L31_Non-Linear Programming Problems - Unconstrained Optimization - KKT Conditions

# L31_Non-Linear Programming Problems - Unconstrained Optimization - KKT Conditions

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CLASSICAL OPTIMIZATION THEORY Quadratic forms Let = n x x x X . . 2 1 be a n-vector.

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Let A = ( a ij ) be a n×n symmetric matrix. We define the k th order principal minor as the k × k determinant kk k k k k a a a a a a a a a . .. . . . .. . .. 2 1 2 22 21 1 12 11
Positive semi definite if X T AX ≥ 0 for all X. Positive definite if X T AX > 0 for all X ≠ 0. Negative semi definite if X T AX ≤ 0 for all X . Negative definite if X T AX < 0 for all X ≠ 0. Then the quadratic form 2 1 2 ii i ij i j i i j n a x a x x ≤ < ≤ = + ∑ ∑ X A X X Q T = ) ( (or the matrix A) is called

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A necessary and sufficient condition for A (or Q ( X )) to be : Negative definite ( negative semi definite ) if k th principal minor of A has the sign of (-1) k , k =1,2,…,n ( k th principal minor of A is zero or has the sign of (-1) k , k =1,2,…,n ) Positive definite ( positive semi definite ) is that all the n principal minors of A are > 0 ( ≥ 0).
Let f ( X )= f ( x 1 , x 2 ,…, x n ) be a real-valued function of the n variables x 1 , x 2 ,…, x n (we assume that f ( X ) is at least twice differentiable ). 0 0 ( ) ( ) f X h f X + r ε j h ) ,.., , ( 2 1 n h h h h = r 0 0 0 0 1 1 2 2 ( , ,.., ) n n X h x h x h x h + = + + + r A point X 0 is said to be a local maximum of f ( X ) if there exists an ε > 0 such that for all Here and 0 0 0 0 1 2 ( , ,.., ) n X x x x =

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X 0 is said to be a local minimum of f ( X ) if there exists an ε > 0 such that X X f X f 2200 ) ( ) ( 0 ) ( ) ( 0 0 X f h X f + r X 0 is called an absolute maximum or global maximum of f ( X ) if   for all X 0 is called an absolute minimum or global minimum of f ( X ) if   0 ( ) ( ) f X f X X 2200 ε j h
Theorem A necessary condition for X 0 to be an optimum point of f ( X ) is that 0 ) ( 0 = X f i x f (that is all the first order partial derivatives are zero at X 0 .) 0 ) ( 0 = X f Definition is called a stationary point of f ( X ) (potential candidate for local maximum or local minimum). A point X 0 for which

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Let X 0 be a stationary point of f ( X ). A sufficient condition for X 0 to be a local minimum of f ( X ) is that the Hessian matrix H ( X 0 ) is positive definite; local maximum of f ( X ) is that the Hessian matrix H ( X 0 ) is negative definite . Theorem
Here H ( X ) is the n×n matrix whose i th row are j f x 2 2 2 2 1 1 2 1 2 2 2 2 1 2 2 2 2 2 2 2 1 2 . . . . . . . . . . n n n n n f f f x x x x x f f f x x x x x f f f x x x x x ∂ ∂ ∂ ∂ ∂ ∂ i.e. H ( X ) = with respect to x i . the partial derivates of ( j =1,2,..n) ( i =1,2,..n)

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Problem 3 set 20.1A page 705 Find the stationary points of the function f ( x 1 , x 2 , x 3 ) = 2 x 1 x 2 x 3 4 x 1 x 3 2 x 2 x 3 + x 1 2 + x 2 2 + x 3 2 2 x 1 - 4 x 2 + 4 x 3 And hence find the extrema of f ( X ) 4 2 2 4 2 4 2 2 2 2 2 4 2 3 2 1 2 1 3 2 3 3 1 2 1 3 3 2 1 + + - - = - + - = - + - = x x x x x x f x x x x x f x x x x
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